MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  kqfeq Structured version   Unicode version

Theorem kqfeq 20519
Description: Two points in the Kolmogorov quotient are equal iff the original points are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
kqval.2  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
Assertion
Ref Expression
kqfeq  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  A. y  e.  J  ( A  e.  y  <->  B  e.  y ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, J, y    x, X, y   
x, V
Allowed substitution hints:    F( x, y)    V( y)

Proof of Theorem kqfeq
StepHypRef Expression
1 kqval.2 . . . . 5  |-  F  =  ( x  e.  X  |->  { y  e.  J  |  x  e.  y } )
21kqfval 20518 . . . 4  |-  ( ( J  e.  V  /\  A  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
323adant3 1019 . . 3  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  A
)  =  { y  e.  J  |  A  e.  y } )
41kqfval 20518 . . . 4  |-  ( ( J  e.  V  /\  B  e.  X )  ->  ( F `  B
)  =  { y  e.  J  |  B  e.  y } )
543adant2 1018 . . 3  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( F `  B
)  =  { y  e.  J  |  B  e.  y } )
63, 5eqeq12d 2426 . 2  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  { y  e.  J  |  A  e.  y }  =  { y  e.  J  |  B  e.  y } ) )
7 rabbi 2988 . 2  |-  ( A. y  e.  J  ( A  e.  y  <->  B  e.  y )  <->  { y  e.  J  |  A  e.  y }  =  {
y  e.  J  |  B  e.  y }
)
86, 7syl6bbr 265 1  |-  ( ( J  e.  V  /\  A  e.  X  /\  B  e.  X )  ->  ( ( F `  A )  =  ( F `  B )  <->  A. y  e.  J  ( A  e.  y  <->  B  e.  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   {crab 2760    |-> cmpt 4455   ` cfv 5571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579
This theorem is referenced by:  ist0-4  20524  kqfvima  20525  kqt0lem  20531  isr0  20532  r0cld  20533  regr1lem2  20535
  Copyright terms: Public domain W3C validator